Determine explicitly the largest disk about the origin whose image under the map

Question

Determine explicitly the largest disk about the origin whose image under the mapping w=z2+z is one to one.

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Explanation / Answer

Consider two points, z1 and z2, mapped to the same point w . Then z1^2 + z1 = z2^2 + z2. Reducing, this implies (z1-z2) (z1+z2+1) = 0 . z1=z2 is not a problem, since this is the definition of 1 to 1. We thus must eliminate the problem of z1+z2+1 = 0, This can only happen if Im(z1) = - Im(z2) and Re(z1) + Real(z2) = -1. In particular, this must work for Im(z) is close to 0, The minimum modulus for which this holds is 1/2. That is, if z1 = -1/2 + ai and z2 = -1/2 -ai, then f(z1) = f(z2), but z1 does not equal z2. If Re(z1) < -1/2, then Re(z2)> -1/2 in order for z1+z1 = -1, so |z2| > 1/2 and that case will be eliminated as long as we restrict ourselves to |z| < 1/2. Thus if z1 = -1/2 + ai and z2 = -1/2 -ai with a > 0 then f(z1) = f(z2) ,and as a goes to zero we have |z1| = |z2| which goes to 1/2 with z1 z2. So, we must have |z| < 1/2. If this holds, then f will be 1 to 1 since |Re(z)| < 1/2 and the condition Re(z1) + Real(z2) = -1 can not hold.

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